DYNAMICAL SYSTEMS
Stampa
Enrollment year
2019/2020
Academic year
2019/2020
Regulations
DM270
Academic discipline
MAT/07 (MATHEMATICAL PHYSICS)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Course
MATHEMATICS
Curriculum
PERCORSO COMUNE
Year of study
Period
2nd semester (02/03/2020 - 09/06/2020)
ECTS
6
Lesson hours
48 lesson hours
Language
English in case of attendance of foreign students
Activity type
ORAL TEST
Teacher
MARZUOLI ANNALISA (titolare) - 6 ECTS
Prerequisites

A course of Analytical Mechanics (Lagrangian and Hamiltonian formulations). Basic knowledge of differential geometry would be helpful.

Learning outcomes

Aim of the course is to make the students acquainted with advanced topics in Analytical Mechanics. A few subjects in the last part of the course will be chosen in agreement with the students’ preferences.

Course contents

Geometrical foundation of Lagrangian and Hamiltonian mechanics. Hamiltonian flux, Liouville’s and Poincaré’s theorems. Symplectic structure on the Hamiltonian phase space; Poincaré-Cartan 1-form and symplectic form. Canonical transformations and their characterization. Algebraic structure of dynamical variables: Poisson brackets and relations with Lie derivatives. Constants of motion and symmetry properties (Hamiltonian Noether’s theorem). Hamilton-Jacobi equations; action-angle variables in the 1-dimensional case and in the n-dimensional, separable case. Completely integrable Hamiltonian systems: Liouville’s and Arnol’d’s theorems. Canonical perturbation theory and an overview of KAM (Kolmogorov, Arnol’d, Moser) theorem. Advanced topics:
i) Canonical perturbation theory and overview of KAM (Kolmogorov, Arnold, Moser) theorem. ii) Poisson manifolds, the method of coadjoint orbits and introduction to geometric quantization; iii) Hamilton-Jacobi theory and semiclassical methods in Quantum Mechanics
Teaching methods

Lectures

Reccomended or required readings

A. Fasano, S. Marmi “Analytical Mechanics: An Introduction”, Oxford University Press 2006

Assessment methods

Oral examination aimed to verify the assimilation of the basic notions and their interconnections.

Further information

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